The Only Maximally Extended, Future-directed, Null and Timelike Geodesics in Gödel Spacetime are Confined to a Submanifold. Drunken Risibility.

Let γ1 be any maximally extended, future-directed, null geodesic confined to a submanifold N whose points all have some particular z ̃ value. Let q be any point in N whose r coordinate satisfies sinh2r = (√2 − 1)/2. Pick any point on γ1. By virtue of the homogeneity of Gödel spacetime, we can find a (temporal orientation preserving) global isometry that maps that point to q and maps N to itself. Let γ2 be the image of γ1 under that isometry. We know that at q the vector (t ̃a + kφa) is null if k = 2(1 + √2). So, by virtue of the isotropy of Gödel spacetime, we can find a global isometry that keeps q fixed, maps N to itself, and rotates γ2 onto a new null geodesic γ3 whose tangent vector at q is, at least, proportional to (t ̃a + 2(1 + √2)φa), with positive proportionality factor. If, finally, we reparametrize γ3 so that its tangent vector at q is equal to (t ̃a + 2(1 + √2)φa), then the resultant curve must be a special null geodesic helix through q since (up to a uniform parameter shift) there can be only one (maximally extended) geodesic through q that has that tangent vector there.

The corresponding argument for timelike geodesics is almost the same. Let γ1 this time be any maximally extended, future-directed, timelike geodesic confined to a submanifold N whose points all have some particular z ̃ value. Let v be the speed of that curve relative to t ̃a. (The value as determined at any point must be constant along the curve since it is a geodesic.). Further, let q be any point in N whose r coordinate satisfies √2(sinh2r)/(cosh2r) = v. (We can certainly find such a point since √2 (sinh2r)/(cosh2r) runs through all values between 0 and 1 as r ranges between 0 and rc/2) Now we can proceed in three stages, as before. We map γ1 to a curve that runs through q. Then we rotate that curve so that its tangent vector (at q) is aligned with (t ̃a + kφa) for the appropriate value of k, namely k = 2 √2/(1 − 2 sinh2r). Finally, we reparametrize the rotated curve so that it has that vector itself as its tangent vector at q. That final curve must be one of our special helical geodesics by the uniqueness theorem for geodesics.

The special timelike and null geodesics we started with – the special helices centered on the axis A – exhibit various features. Some are exhibited by all timelike and null geodesics (confined to a z ̃ = constant submanifold); some are not. It is important to keep track of the difference. What is at issue is whether the features can or cannot be captured in terms of gab, t ̃a, and z ̃a (or whether they make essential reference to the coordinates t ̃, r, φ themselves). So, for example, if a curve is parametrized by s, one might take its vertical “pitch” (relative to t ̃) at any point to be given by the value of dt ̃/ds there. Understood this way, the vertical pitch of the special helices centered on A is constant, but that of other timelike and null geodesics is not. For this reason, it is not correct to think of the latter, simply, as “translated” versions of the former. On the other hand, the following is true of all timelike and null geodesics (confined to a z ̃ = constant submanifold). If we project them (via t ̃a) onto a two-dimensional submanifold characterized by constant values for t ̃ as well as z ̃, the result is a circle.

Here is another way to make the point. Consider any timelike or null geodesic γ (confined to a z ̃ = constant submanifold). It certainly need not be centered on the axis A and need not have constant vertical pitch relative to t ̃. But we can always find a (new) axis A′ and a new set of cylindrical coordinates t ̃′, r′, φ′ adapted to A′ such that γ qualifies as a special helical geodesic relative to those coordinates. In particular, it will have constant vertical pitch relative to t ̃′.

Let us now consider all the timelike and null geodesics that pass through some point p (and are confined to a z ̃ = constant submanifold). It may as well be on the original axis A. We can better visualize the possibilities if we direct our attention to the circles that arise after projection (via t ̃a). The figure below shows a two-dimensional submanifold through p on which t ̃ and z ̃ are both constant. The dotted circle has radius rc. Once again, that is the “critical radius” at which the rotational Killing field φa is null. Call this dotted circle the “critical circle.” The circles that pass through p and have radius r = rc/2 are projections of null geodesics. Each shares exactly one point with the critical circle. In contrast, the circles of smaller radius that pass through p are the projections of timelike geodesics. The figure captures one of the claims – namely, that no timelike or null geodesic that passes through a point can “escape” to a radial distance from it greater than rc.

Untitled

Figure: Projections of timelike and null geodesics in Gödel spacetime. rc is the “critical radius” at which the rotational Killing field φa centered at p is null

Gödel spacetime exhibits a “boomerang effect.” Suppose an individual is at rest with respect to the cosmic source fluid in Gödel spacetime (and so his worldline coincides with some t ̃-line). If that individual shoots a gun at some point, in any direction orthogonal to z ̃a, then, no matter what the muzzle speed of the gun, the bullet will eventually come back and hit him (unless it hits something else first or disintegrates).

Killing Fields

Let κa be a smooth field on our background spacetime (M, gab). κa is said to be a Killing field if its associated local flow maps Γs are all isometries or, equivalently, if £κ gab = 0. The latter condition can also be expressed as ∇(aκb) = 0.

Any number of standard symmetry conditions—local versions of them, at least can be cast as claims about the existence of Killing fields. Local, because killing fields need not be complete, and their associated flow maps need not be defined globally.

(M, gab) is stationary if it has a Killing field that is everywhere timelike.

(M, gab) is static if it has a Killing field that is everywhere timelike and locally hypersurface orthogonal.

(M, gab) is homogeneous if its Killing fields, at every point of M, span the tangent space.

In a stationary spacetime there is, at least locally, a “timelike flow” that preserves all spacetime distances. But the flow can exhibit rotation. Think of a whirlpool. It is the latter possibility that is ruled out when one passes to a static spacetime. For example, Gödel spacetime, is stationary but not static.

Let κa be a Killing field in an arbitrary spacetime (M, gab) (not necessarily Minkowski spacetime), and let γ : I → M be a smooth, future-directed, timelike curve, with unit tangent field ξa. We take its image to represent the worldline of a point particle with mass m > 0. Consider the quantity J = (Paκa), where Pa = mξa is the four-momentum of the particle. It certainly need not be constant on γ[I]. But it will be if γ is a geodesic. For in that case, ξnnξa = 0 and hence

ξnnJ = m(κa ξnnξa + ξnξanκa) = mξnξa ∇(nκa) = 0

Thus, J is constant along the worldlines of free particles of positive mass. We refer to J as the conserved quantity associated with κa. If κa is timelike, we call J the energy of the particle (associated with κa). If it is spacelike, and if its associated flow maps resemble translations, we call J the linear momentum of the particle (associated with κa). Finally, if κa is spacelike, and if its associated flow maps resemble rotations, then we call J the angular momentum of the particle (associated with κa).

It is useful to keep in mind a certain picture that helps one “see” why the angular momentum of free particles (to take that example) is conserved. It involves an analogue of angular momentum in Euclidean plane geometry. Figure below shows a rotational Killing field κa in the Euclidean plane, the image of a geodesic (i.e., a line) L, and the tangent field ξa to the geodesic. Consider the quantity J = ξaκa, i.e., the inner product of ξa with κa – along L, and we can better visualize the assertion.

Untitled

Figure: κa is a rotational Killing field. (It is everywhere orthogonal to a circle radius, and is proportional to it in length.) ξa is a tangent vector field of constant length on the line L. The inner product between them is constant. (Equivalently, the length of the projection of κa onto the line is constant.)

Let us temporarily drop indices and write κ·ξ as one would in ordinary Euclidean vector calculus (rather than ξaκa). Let p be the point on L that is closest to the center point where κ vanishes. At that point, κ is parallel to ξ. As one moves away from p along L, in either direction, the length ∥κ∥ of κ grows, but the angle ∠(κ,ξ) between the vectors increases as well. It should seem at least plausible from the picture that the length of the projection of κ onto the line is constant and, hence, that the inner product κ·ξ = cos(∠(κ , ξ )) ∥κ ∥ ∥ξ ∥ is constant.

That is how to think about the conservation of angular momentum for free particles in relativity theory. It does not matter that in the latter context we are dealing with a Lorentzian metric and allowing for curvature. The claim is still that a certain inner product of vector fields remains constant along a geodesic, and we can still think of that constancy as arising from a compensatory balance of two factors.

Let us now turn to the second type of conserved quantity, the one that is an attribute of extended bodies. Let κa be an arbitrary Killing field, and let Tab be the energy-momentum field associated with some matter field. Assume it satisfies the conservation condition (∇aTab = 0). Then (Tabκb) is divergence free:

a(Tabκb) = κbaTab + Tabaκb = Tab∇(aκb) = 0

(The second equality follows from the conservation condition and the symmetry of Tab; the third follows from the fact that κa is a Killing field.) It is natural, then, to apply Stokes’s theorem to the vector field (Tabκb). Consider a bounded system with aggregate energy-momentum field Tab in an otherwise empty universe. Then there exists a (possibly huge) timelike world tube such that Tab vanishes outside the tube (and vanishes on its boundary).

Let S1 and S2 be (non-intersecting) spacelike hypersurfaces that cut the tube as in the figure below, and let N be the segment of the tube falling between them (with boundaries included).

Untitled

Figure: The integrated energy (relative to a background timelike Killing field) over the intersection of the world tube with a spacelike hypersurface is independent of the choice of hypersurface.

By Stokes’s theorem,

S2(Tabκb)dSa – ∫S1(Tabκb)dSa = ∫S2∩∂N(Tabκb)dSa – ∫S1∩∂N(Tabκb)dSa

= ∫∂N(Tabκb)dSa = ∫Na(Tabκb)dV = 0

Thus, the integral ∫S(Tabκb)dSa is independent of the choice of spacelike hypersurface S intersecting the world tube, and is, in this sense, a conserved quantity (construed as an attribute of the system confined to the tube). An “early” intersection yields the same value as a “late” one. Again, the character of the background Killing field κa determines our description of the conserved quantity in question. If κa is timelike, we take ∫S(Tabκb)dSa to be the aggregate energy of the system (associated with κa). And so forth.

Magnetic Field as the Rotational Component of Electromagnetic Field

Let (M, gab) be the background relativistic spacetime. We are assuming it is temporally orientable and endowed with a particular temporal orientation. Let ξa be a smooth, future-directed unit timelike vector field on M (or some open subset of M). We understand it to represent the four-velocity field of a fluid. Further, let hab be the spatial projection field determined by ξa. The rotation and expansion fields associated with ξa are defined as follows:

ωab = h[amhb]nmξn —– (1)

θab = h(amhb)nmξn —– (2)

They are smooth fields, orthogonal to ξa in both indices, and satisfy

aξb = ωab + θab + ξammξb) —– (3)

Let γ be an integral curve of ξa, and let p be a point on the image of γ. Further, let ηa be a vector field on the image of γ that is carried along by the flow of ξa and orthogonal to ξa at p. (It need not be orthogonal to ξa elsewhere.) We think of the image of γ as the worldline of a fluid element O, and think of ηa at p as a “connecting vector” that spans the distance between O and a neighboring fluid element N that is “infinitesimally close.” The instantaneous velocity of N relative to O at p is given by ξaaηb. But ξaaηb = ηaaξb. So, by equation (3) and the orthogonality of ξa with ηa at p, we have

ξaaηb = (ωab + θaba —– (4)

at the point. Here we have simply decomposed the relative velocity vector into two components. The first, (ωabηa), is orthogonal to ηa since ωab is anti-symmetric. It is naturally understood as the instantaneous rotational velocity of N with respect to O at p.

Untitled

The angular velocity (or twist) vector ωa. It points in the direction of the instantaneous axis of rotation of the fluid. Its magnitude ∥ωa∥ is the instantaneous angular speed of the fluid about that axis. Here ηa connects the fluid element O to the “infinitesimally close” fluid element N. The rotational velocity of N relative to O is given by ωbaηb. The latter is orthogonal to ηa

In support of this interpretation, consider the instantaneous rate of change of the squared length (−ηbηb) of ηa at p. It follows from equation (4) that

ξaa(−ηbηb) = −2θabηaηb —– (5)

Thus the rate of change depends solely on θab. Suppose θab = 0. Then the instantaneous velocity of N with respect to O at p has a vanishing radial component. If ωab ≠ 0, N can still have non-zero velocity there with respect to O. But it can only be a rotational velocity. The two conditions (θab = 0 and ωab ≠ 0) jointly characterize “rigid rotation.”

The rotation tensor ωab at a point p determines both an (instantaneous) axis of rotation there, and an (instantaneous) speed of rotation. As we shall see, both pieces of information are built into the angular velocity (or twist) vector

ωa = 1/2 εabcd ξbωcd —– (6)

at p. (Here εabcd is a volume element defined on some open set containing p. Clearly, if we switched from the volume element εabcd to its negation, the result would be to replace ωa with −ωa.)

If follows from equation (6) (and the anti-symmetry of εabcd) that ωa is orthogonal to ξa. It further follows that

ωa = 1/2 εabcd ξbcξd —– (7)

ωab = εabcd ξcωd —– (8)

Hence, ωab = 0 iff ωa = 0.

a = εabcd ξbωcd = εabcd ξb h[crhd]srξs = εabcd ξbhcrhdsrξ

= εabcd ξbgcr gdsrξs = εabcd ξbcξd

The second equality follows from the anti-symmetry of εabcd, and the third from the fact that εabcdξb is orthogonal to ξa in all indices.) The equation (6) has exactly the same form as the definition of the magnetic field vector Ba determined relative to a Maxwell field Fab and four-velocity vector ξa (Ba = 1/2 εabcd ξb Fcd). It is for this reason that the magnetic field is sometimes described as the “rotational component of the electromagnetic field.”

Dynamics of Point Particles: Orthogonality and Proportionality

optical

Let γ be a smooth, future-directed, timelike curve with unit tangent field ξa in our background spacetime (M, gab). We suppose that some massive point particle O has (the image of) this curve as its worldline. Further, let p be a point on the image of γ and let λa be a vector at p. Then there is a natural decomposition of λa into components proportional to, and orthogonal to, ξa:

λa = (λbξba + (λa −(λbξba) —– (1)

Here, the first part of the sum is proportional to ξa, whereas the second one is orthogonal to ξa.

These are standardly interpreted, respectively, as the “temporal” and “spatial” components of λa relative to ξa (or relative to O). In particular, the three-dimensional vector space of vectors at p orthogonal to ξa is interpreted as the “infinitesimal” simultaneity slice of O at p. If we introduce the tangent and orthogonal projection operators

kab = ξa ξb —– (2)

hab = gab − ξa ξb —– (3)

then the decomposition can be expressed in the form

λa = kab λb + hab λb —– (4)

We can think of kab and hab as the relative temporal and spatial metrics determined by ξa. They are symmetric and satisfy

kabkbc = kac —– (5)

habhbc = hac —– (6)

Many standard textbook assertions concerning the kinematics and dynamics of point particles can be recovered using these decomposition formulas. For example, suppose that the worldline of a second particle O′ also passes through p and that its four-velocity at p is ξ′a. (Since ξa and ξ′a are both future-directed, they are co-oriented; i.e., ξa ξ′a > 0.) We compute the speed of O′ as determined by O. To do so, we take the spatial magnitude of ξ′a relative to O and divide by its temporal magnitude relative to O:

v = speed of O′ relative to O = ∥hab ξ′b∥ / ∥kab ξ′b∥ —– (7)

For any vector μa, ∥μa∥ is (μaμa)1/2 if μ is causal, and it is (−μaμa)1/2 otherwise.

We have from equations 2, 3, 5 and 6

∥kab ξ′b∥ = (kab ξ′b kac ξ′c)1/2 = (kbc ξ′bξ′c)1/2 = (ξ′bξb)

and

∥hab ξ′b∥ = (−hab ξ′b hac ξ′c)1/2 = (−hbc ξ′bξ′c)1/2 = ((ξ′bξb)2 − 1)1/2

so

v = ((ξ’bξb)2 − 1)1/2 / (ξ′bξb) < 1 —– (8)

Thus, as measured by O, no massive particle can ever attain the maximal speed 1. We note that equation (8) implies that

(ξ′bξb) = 1/√(1 – v2) —– (9)

It is a basic fact of relativistic life that there is associated with every point particle, at every event on its worldline, a four-momentum (or energy-momentum) vector Pa that is tangent to its worldline there. The length ∥Pa∥ of this vector is what we would otherwise call the mass (or inertial mass or rest mass) of the particle. So, in particular, if Pa is timelike, we can write it in the form Pa =mξa, where m = ∥Pa∥ > 0 and ξa is the four-velocity of the particle. No such decomposition is possible when Pa is null and m = ∥Pa∥ = 0.

Suppose a particle O with positive mass has four-velocity ξa at a point, and another particle O′ has four-momentum Pa there. The latter can either be a particle with positive mass or mass 0. We can recover the usual expressions for the energy and three-momentum of the second particle relative to O if we decompose Pa in terms of ξa. By equations (4) and (2), we have

Pa = (Pbξb) ξa + habPb —– (10)

the first part of the sum is the energy component, while the second is the three-momentum. The energy relative to O is the coefficient in the first term: E = Pbξb. If O′ has positive mass and Pa = mξ′a, this yields, by equation (9),

E = m (ξ′bξb) = m/√(1 − v2) —– (11)

(If we had not chosen units in which c = 1, the numerator in the final expression would have been mc2 and the denominator √(1 − (v2/c2)). The three−momentum relative to O is the second term habPb in the decomposition of Pa, i.e., the component of Pa orthogonal to ξa. It follows from equations (8) and (9) that it has magnitude

p = ∥hab mξ′b∥ = m((ξ′bξb)2 − 1)1/2 = mv/√(1 − v2) —– (12)

Interpretive principle asserts that the worldlines of free particles with positive mass are the images of timelike geodesics. It can be thought of as a relativistic version of Newton’s first law of motion. Now we consider acceleration and a relativistic version of the second law. Once again, let γ : I → M be a smooth, future-directed, timelike curve with unit tangent field ξa. Just as we understand ξa to be the four-velocity field of a massive point particle (that has the image of γ as its worldline), so we understand ξnnξa – the directional derivative of ξa in the direction ξa – to be its four-acceleration field (or just acceleration) field). The four-acceleration vector at any point is orthogonal to ξa. (This is, since ξannξa) = 1/2 ξnnaξa) = 1/2 ξnn (1) = 0). The magnitude ∥ξnnξa∥ of the four-acceleration vector at a point is just what we would otherwise describe as the curvature of γ there. It is a measure of the rate at which γ “changes direction.” (And γ is a geodesic precisely if its curvature vanishes everywhere).

The notion of spacetime acceleration requires attention. Consider an example. Suppose you decide to end it all and jump off the tower. What would your acceleration history be like during your final moments? One is accustomed in such cases to think in terms of acceleration relative to the earth. So one would say that you undergo acceleration between the time of your jump and your calamitous arrival. But on the present account, that description has things backwards. Between jump and arrival, you are not accelerating. You are in a state of free fall and moving (approximately) along a spacetime geodesic. But before the jump, and after the arrival, you are accelerating. The floor of the observation deck, and then later the sidewalk, push you away from a geodesic path. The all-important idea here is that we are incorporating the “gravitational field” into the geometric structure of spacetime, and particles traverse geodesics iff they are acted on by no forces “except gravity.”

The acceleration of our massive point particle – i.e., its deviation from a geodesic trajectory – is determined by the forces acting on it (other than “gravity”). If it has mass m, and if the vector field Fa on I represents the vector sum of the various (non-gravitational) forces acting on it, then the particle’s four-acceleration ξnnξa satisfies

Fa = mξnnξa —– (13)

This is Newton’s second law of motion. Consider an example. Electromagnetic fields are represented by smooth, anti-symmetric fields Fab. If a particle with mass m > 0, charge q, and four-velocity field ξa is present, the force exerted by the field on the particle at a point is given by qFabξb. If we use this expression for the left side of equation (13), we arrive at the Lorentz law of motion for charged particles in the presence of an electromagnetic field:

qFabξb = mξbbξa —– (14)

This equation makes geometric sense. The acceleration field on the right is orthogonal to ξa. But so is the force field on the left, since ξa(Fabξb) = ξaξbFab = ξaξbF(ab), and F(ab) = 0 by the anti-symmetry of Fab.

Bernard Cache’s Earth Moves: The Furnishing of Territories (Writing Architecture)

bernard_cache_lectur

Take the concept of singularity. In mathematics, what is said to be singular is not a given point, but rather a set of points on a given curve. A point is not singular; it becomes singularized on a continuum. And several types of singularity exist, starting with fractures in curves and other bumps in the road. We will discount them at the outset, for singularities that are marked by discontinuity signal events that are exterior to the curvature and are themselves easily identifiable. In the same way, we will eliminate singularities such as backup points [points de rebroussement]. For though they are indeed discontinuous, they refer to a vector that is tangential to the curve and thus trace a symmetrical axis that constitutive of the backup point. Whether it be a reflection of the tan- gential plane or a rebound with respect to the orthogonal plane, the backup point is thus not a basic singularity. It is rather the result of an operation effectuated on any part of the curve. Here again, the singular would be the sign of too noisy, too memorable an event, while what we want to do is to deal with what is most smooth: ordinary continua, sleek and polished.

On one hand there are the extrema, the maximum and minimum on a given curve. And on the other there are those singular points that, in relation to the extrema, figure as in-betweens. These are known as points of inflection. They are different from the extrema in that they are defined only in relation to themselves, whereas the definition of the extrema presupposes the prior choice of an axis or an orientation, that is to say of a vector.

Indeed, a maximum or a minimum is a point where the tangent to the curve is directed perpendicularly to the axis of the ordinates [y-axis]. Any new orientation of the coordinate axes repositions the maxima and the min- ima; they are thus extrinsic singularities. The point of inflection, however, designates a pure event of curvature where the tangent crosses the curve; yet this event does not depend in any way on the orientation of the axes, which is why it can be said that inflection is an intrinsic singularity. On either side of the inflection, we know that there will be a highest point and a lowest point, but we cannot designate them as long as the curve has not been related to the orientation of a vector. Points of inflection are singularities in and of themselves, while they confer an indeterminacy to the rest of the curve. Preceding the vector, inflection makes of each of the points a possible extremum in relation to its inverse: virtual maxima and minima. In this way, inflection represents a totality of possibilities, as well as an openness, a receptiveness, or an anticipation……

Bernard Cache Earth Moves The Furnishing of Territories

Holonomies: Philosophies of Conjugacy. Part 1.

Figure-6-Holonomy-along-a-leafwise-path

Suppose that N is an irreducible 2n-dimensional Riemannian symmetric space. We may realise N as a coset space N = G/K with Gτ ⊂ K ⊂ (Gτ)0 for some involution τ of G. Now K is (a covering of) the holonomy group of N and similarly the coset fibration G → G/K covers the holonomy bundle P → N. In this setting, J(N) is associated to G:

J(N) ≅ G ×K J (R2n)

and if K/H is a K-orbit in J(R2n) then the corresponding subbundle is G ×K K/H = G/H and the projection is just the coset fibration. Thus, the subbundles of J(N) are just the orbits of G in J(N).

Let j ∈ J (N). Then G · j is an almost complex submanifold of J (N) on which J is integrable iff j lies in the zero-set of the Nijenhuis tensor NJ.

This focusses our attention on the zero-set of NJ which we denote by Z. In favourable circumstances, the structure of this set can be completely described. We begin by assuming that N is of compact type so that G is compact and semi-simple. We also assume that N is inner i.e. that τ is an inner involution of G or, equivalently, that rankG = rankK. The class of inner symmetric spaces include the even-dimensional spheres, the Hermitian symmetric spaces, the quaternionic Kähler symmetric spaces and indeed all symmetric G-spaces for G = SO(2n+1), Sp(n), E7, E8, F4 and G2. Moreover, all inner symmetric spaces are necessarily even-dimensional and so fit into our framework.

Let N = G/K be a simply-connected inner Riemannian symmetric space of compact type. Then Z consists of a finite number of connected components on each of which G acts transitively. Moreover, any G-flag manifold is realised as such an orbit for some N.

The proof for the above requires a detour into the geometry of flag manifolds and reveals an interesting interaction between the complex geometry of flag manifolds and the real geometry of inner symmetric spaces. For this, we begin by noting that a coset space of the form G/C(T) admits several invariant Kählerian complex structures in general. Using a complex realisation of G/C(T) as follows: having fixed a complex structure, the complexified group GC acts transitively on G/C(T) by biholomorphisms with parabolic subgroups as stabilisers. Conversely, if P ⊂ GC is a parabolic subgroup then the action of G on GC/P is transitive and G ∩ P is the centraliser of a torus in G. For the infinitesimal situation: let F = G/C(T) be a flag manifold and let o ∈ F. We have a splitting of the Lie algebra of G

gC = h ⊕ m

with m ≅ ToF and h the Lie algebra of the stabiliser of o in G. An invariant complex structure on F induces an ad h-invariant splitting of mC into (1, 0) and (0, 1) spaces mC = m+ ⊕ m− with [m+, m+] ⊂ m+ by integrability. One can show that m+ and m are nilpotent subalgebras of gC and in fact hC ⊕ m is a parabolic subalgebra of gC with nilradical m. If P is the corresponding parabolic subgroup of GC then P is the stabiliser of o and we obtain a biholomorphism between the complex coset space GC/P and the flag manifold F.

Conversely, let P ⊂ GC be a parabolic subgroup with Lie algebra p and let n be the conjugate of the nilradical of p (with respect to the real form g). Then H = G ∩ P is the centraliser of a torus and we have orthogonal decompositions (with respect to the Killing inner product)

p = hC ⊕ n, gC = hC ⊕ n ⊕ n

which define an invariant complex structure on G/H realising the biholomorphism with GC/P.

The relationship between a flag manifold F = GC/P and an inner symmetric space comes from an examination of the central descending series of n. This is a filtration 0 = nk+1 ⊂ nk ⊂…⊂ n1 = n of n defined by ni = [n, ni−1].

We orthogonalise this filtration using the Killing inner product by setting

gi = ni+1 ∩ ni

for i ≥ 1 and extend this to a decomposition of gC by setting g0 = hC = (g ∩ p)C and g−i = gfor i ≥ 1. Then

gC = ∑gi

is an orthogonal decomposition with

p = ∑i≤0 gi, n = ∑i>0 g

The crucial property of this decomposition is that

[gi, gj] ⊂ gi+j

which can be proved by demonstrating the existence of an element ξ ∈ h with the property that, for each i, adξ has eigenvalue √−1i on gi. This element ξ (necessarily unique since g is semi-simple) is the canonical element of p. Since ad ξ has eigenvalues in √−1Z, ad exp πξ is an involution of g which we exponentiate to obtain an inner involution τξ of G and thus an inner symmetric space G/K where K = (Gτξ)0. Clearly, K has Lie algebra given by

k = g ∩ ∑i g2i

Conjuncted: Twistor Spaces

scan0003

The α-planes can be generalized to a suitable class of curved complex space-times. By a complex space-time (M,g) we mean a four-dimensional Hausdorff manifold M with holomorphic metric g. Thus, with respect to a holomorphic coordinate basis xa, g is a 4×4 matrix of holomorphic functions of xa, and its determinant is nowhere-vanishing. Remarkably, g determines a unique holomorphic connection ∇, and a holomorphic curvature tensor Rabcd. Moreover, the Ricci tensor Rab becomes complex-valued, and the Weyl tensor Cabcd may be split into independent holomorphic tensors, i.e. its self-dual and anti-self-dual parts, respectively. With our two-spinor notation, one has

Cabcd = ψABCD εA′B′ εC′D′ + ψ~A′B′C′D′ εAB εCD

where ψABCD = ψ(ABCD), ψA′B′C′D′ = ψ~(A′B′C′D′). The spinors ψ and ψ~ are the anti-self-dual and self-dual Weyl spinors, respectively.

ψ~A′B′C′D′ = 0, Rab = 0, are called right-flat or anti-self-dual, whereas complex vacuum space-times such that

ψABCD = 0, Rab = 0,

are called left-flat or self-dual. This definition only makes sense if space-time is complex or real Riemannian, since in this case no complex conjugation relates primed to unprimed spinors (i.e. the corresponding spin-spaces are no longer anti-isomorphic). Hence, for example, the self-dual Weyl spinor ψ~A′B′C′D′ may vanish without its anti-self-dual counterpart ψABCD having to vanish as well, or the converse may hold.

By definition, α-surfaces are complex two-surfaces S in a complex space-time (M, g) whose tangent vectors v have the two-spinor form, where λA is varying, and πA is a fixed primed spinor field on S. From this definition, the following properties can be derived:

(i) tangent vectors to α-surfaces are null;

(ii) any two null tangent vectors v and u to an α-surface are orthogonal to one another;

(iii) the holomorphic metric g vanishes on S in that g(v, u) = g(v, v) = 0, ∀ v, u, so that α-surfaces are totally null;

(iv) α-surfaces are self-dual, in that F ≡ v ⊗ u − u ⊗ v takes the two-spinor form;

(v) α-surfaces exist in (M,g) iff the self-dual Weyl spinor vanishes, so that (M, g) is anti-self-dual.

The properties (i)–(iv), are the same as in the flat-space-time case, provided we replace the flat metric η with the curved metric g. Condition (v), however, is a peculiarity of curved space-times.

We want to prove that, if (M,g) is anti-self-dual, it admits a three-complex- parameter family of self-dual α-surfaces. Indeed, given any point p ∈ M and a spinor μA at p, one can find a spinor field πA on M, satisfying the equation

πAAA πB = ξAπB,

and such that

πA(p) = μA(p)

Hence πA defines a holomorphic two-dimensional distribution, spanned by the vector fields of the form λAπA, which is integrable. Thus, in particular, there exists a self-dual α-surface through p, with tangent vectors of the form λAμA at p. Since p is arbitrary, this argument may be repeated ∀p ∈ M. The space P of all self-dual α-surfaces in (M,g) is three-complex-dimensional, and is called twistor space of (M, g).

I15-56-trouser

Extreme Value Theory

1469941517622

Standard estimators of the dependence between assets are the correlation coefficient or the Spearman’s rank correlation for instance. However, as stressed by [Embrechts et al. ], these kind of dependence measures suffer from many deficiencies. Moreoever, their values are mostly controlled by relatively small moves of the asset prices around their mean. To cure this problem, it has been proposed to use the correlation coefficients conditioned on large movements of the assets. But [Boyer et al.] have emphasized that this approach suffers also from a severe systematic bias leading to spurious strategies: the conditional correlation in general evolves with time even when the true non-conditional correlation remains constant. In fact, [Malevergne and Sornette] have shown that any approach based on conditional dependence measures implies a spurious change of the intrinsic value of the dependence, measured for instance by copulas. Recall that the copula of several random variables is the (unique) function which completely embodies the dependence between these variables, irrespective of their marginal behavior (see [Nelsen] for a mathematical description of the notion of copula).

In view of these limitations of the standard statistical tools, it is natural to turn to extreme value theory. In the univariate case, extreme value theory is very useful and provides many tools for investigating the extreme tails of distributions of assets returns. These new developments rest on the existence of a few fundamental results on extremes, such as the Gnedenko-Pickands-Balkema-de Haan theorem which gives a general expression for the distribution of exceedence over a large threshold. In this framework, the study of large and extreme co-movements requires the multivariate extreme values theory, which unfortunately does not provide strong results. Indeed, in constrast with the univariate case, the class of limiting extreme-value distributions is too broad and cannot be used to constrain accurately the distribution of large co-movements.

In the spirit of the mean-variance portfolio or of utility theory which establish an investment decision on a unique risk measure, we use the coefficient of tail dependence, which, to our knowledge, was first introduced in the financial context by [Embrechts et al.]. The coefficient of tail dependence between assets Xi and Xj is a very natural and easy to understand measure of extreme co-movements. It is defined as the probability that the asset Xi incurs a large loss (or gain) assuming that the asset Xj also undergoes a large loss (or gain) at the same probability level, in the limit where this probability level explores the extreme tails of the distribution of returns of the two assets. Mathematically speaking, the coefficient of lower tail dependence between the two assets Xi and Xj , denoted by λ−ij is defined by

λ−ij = limu→0 Pr{Xi<Fi−1(u)|Xj < Fj−1(u)} —– (1)

where Fi−1(u) and Fj−1(u) represent the quantiles of assets Xand Xj at level u. Similarly the coefficient of the upper tail dependence is

λ+ij = limu→1 Pr{Xi > Fi−1(u)|Xj > Fj−1(u)} —– (2)

λ−ij and λ+ij are of concern to investors with long (respectively short) positions. We refer to [Coles et al.] and references therein for a survey of the properties of the coefficient of tail dependence. Let us stress that the use of quantiles in the definition of λ−ij and λ+ij makes them independent of the marginal distribution of the asset returns: as a consequence, the tail dependence parameters are intrinsic dependence measures. The obvious gain is an “orthogonal” decomposition of the risks into (1) individual risks carried by the marginal distribution of each asset and (2) their collective risk described by their dependence structure or copula.

Being a probability, the coefficient of tail dependence varies between 0 and 1. A large value of λ−ij means that large losses occur almost surely together. Then, large risks can not be diversified away and the assets crash together. This investor and portfolio manager nightmare is further amplified in real life situations by the limited liquidity of markets. When λ−ij vanishes, these assets are said to be asymptotically independent, but this term hides the subtlety that the assets can still present a non-zero dependence in their tails. For instance, two normally distributed assets can be shown to have a vanishing coefficient of tail dependence. Nevertheless, unless their correlation coefficient is identically zero, these assets are never independent. Thus, asymptotic independence must be understood as the weakest dependence which can be quantified by the coefficient of tail dependence.

For practical implementations, a direct application of the definitions (1) and (2) fails to provide reasonable estimations due to the double curse of dimensionality and undersampling of extreme values, so that a fully non-parametric approach is not reliable. It turns out to be possible to circumvent this fundamental difficulty by considering the general class of factor models, which are among the most widespread and versatile models in finance. They come in two classes: multiplicative and additive factor models respectively. The multiplicative factor models are generally used to model asset fluctuations due to an underlying stochastic volatility for a survey of the properties of these models). The additive factor models are made to relate asset fluctuations to market fluctuations, as in the Capital Asset Pricing Model (CAPM) and its generalizations, or to any set of common factors as in Arbitrage Pricing Theory. The coefficient of tail dependence is known in close form for both classes of factor models, which allows for an efficient empirical estimation.

Purely Random Correlations of the Matrix, or Studying Noise in Neural Networks

2-Figure1-1

Expressed in the most general form, in essentially all the cases of practical interest, the n × n matrices W used to describe the complex system are by construction designed as

W = XYT —– (1)

where X and Y denote the rectangular n × m matrices. Such, for instance, are the correlation matrices whose standard form corresponds to Y = X. In this case one thinks of n observations or cases, each represented by a m dimensional row vector xi (yi), (i = 1, …, n), and typically m is larger than n. In the limit of purely random correlations the matrix W is then said to be a Wishart matrix. The resulting density ρW(λ) of eigenvalues is here known analytically, with the limits (λmin ≤ λ ≤ λmax) prescribed by

λmaxmin = 1+1/Q±2 1/Q and Q = m/n ≥ 1.

The variance of the elements of xi is here assumed unity.

The more general case, of X and Y different, results in asymmetric correlation matrices with complex eigenvalues λ. In this more general case a limiting distribution corresponding to purely random correlations seems not to be yet known analytically as a function of m/n. It indicates however that in the case of no correlations, quite generically, one may expect a largely uniform distribution of λ bound in an ellipse on the complex plane.

Further examples of matrices of similar structure, of great interest from the point of view of complexity, include the Hamiltonian matrices of strongly interacting quantum many body systems such as atomic nuclei. This holds true on the level of bound states where the problem is described by the Hermitian matrices, as well as for excitations embedded in the continuum. This later case can be formulated in terms of an open quantum system, which is represented by a complex non-Hermitian Hamiltonian matrix. Several neural network models also belong to this category of matrix structure. In this domain the reference is provided by the Gaussian (orthogonal, unitary, symplectic) ensembles of random matrices with the semi-circle law for the eigenvalue distribution. For the irreversible processes there exists their complex version with a special case, the so-called scattering ensemble, which accounts for S-matrix unitarity.

As it has already been expressed above, several variants of ensembles of the random matrices provide an appropriate and natural reference for quantifying various characteristics of complexity. The bulk of such characteristics is expected to be consistent with Random Matrix Theory (RMT), and in fact there exists strong evidence that it is. Once this is established, even more interesting are however deviations, especially those signaling emergence of synchronous or coherent patterns, i.e., the effects connected with the reduction of dimensionality. In the matrix terminology such patterns can thus be associated with a significantly reduced rank k (thus k ≪ n) of a leading component of W. A satisfactory structure of the matrix that would allow some coexistence of chaos or noise and of collectivity thus reads:

W = Wr + Wc —– (2)

Of course, in the absence of Wr, the second term (Wc) of W generates k nonzero eigenvalues, and all the remaining ones (n − k) constitute the zero modes. When Wr enters as a noise (random like matrix) correction, a trace of the above effect is expected to remain, i.e., k large eigenvalues and the bulk composed of n − k small eigenvalues whose distribution and fluctuations are consistent with an appropriate version of random matrix ensemble. One likely mechanism that may lead to such a segregation of eigenspectra is that m in eq. (1) is significantly smaller than n, or that the number of large components makes it effectively small on the level of large entries w of W. Such an effective reduction of m (M = meff) is then expressed by the following distribution P(w) of the large off-diagonal matrix elements in the case they are still generated by the random like processes

P(w) = (|w|(M-1)/2K(M-1)/2(|w|))/(2(M-1)/2Γ(M/2)√π) —– (3)

where K stands for the modified Bessel function. Asymptotically, for large w, this leads to P(w) ∼ e(−|w|) |w|M/2−1, and thus reflects an enhanced probability of appearence of a few large off-diagonal matrix elements as compared to a Gaussian distribution. As consistent with the central limit theorem the distribution quickly converges to a Gaussian with increasing M.

Based on several examples of natural complex dynamical systems, like the strongly interacting Fermi systems, the human brain and the financial markets, one could systematize evidence that such effects are indeed common to all the phenomena that intuitively can be qualified as complex.

Roger Penrose and Artificial Intelligence: Revenance from the Archives and the Archaic.

Let us have a look at Penrose and his criticisms of strong AI, and does he come out as a winner. His Emperor’s New Mind: Concerning Computers, Minds, and The Laws of Physics

sets out to deal a death blow to the project of strong AI. Even while showing humility, like in saying,

My point of view is an unconventional among physicists and is consequently one which is unlikely to be adopted, at present, by computer scientists or physiologists,

he is merely stressing on his speculative musings. Penrosian arguments ala Searle, are definitely opinionated, in making assertions like a conscious mind cannot work like a computer. He grants the possibility of artificial machines coming into existence, and even superseding humans (1), but at every moment remains convinced that algorithmic machines are doomed to subservience. Penrose’s arguments proceed through showing that human intelligence cannot be implemented by any Turing machine equivalent computer, and human mind as not algorithmically based that could capture the Turing machine equivalent. He is even sympathetic to Searle’s Chinese Room argument, despite showing some reservations against its conclusions.

chinese-room2

The speculative nature of his arguments question people as devices which compute that a Turing machine cannot, despite physical laws that allow such a construction of a device as a difficult venture. This is where his quantum theory sets in, with U and R (Unitary process and Reduction process respectively) acting on quantum states that help describe a quantum system. His U and R processes and the states they act upon are not only independent of observers, but at the same time real, thus branding him as a realist. What happens is an interpolation that occurs between Unitary Process and Reductive Process, a new procedure that essentially contains a non-algorithmic element takes shape, which effectuates a future that cannot be computable based on the present, even though it could be determined that way. This radically new concept which is applied to space-time is mapped onto the depths of brain’s structure, and for Penrose, the speculative possibility occurs in what he terms the Phenomenon of Brain Plasticity. As he says,

Somewhere within the depths of the brain, as yet unknown cells are to be found of single quantum sensitivity, such that synapses becoming activate or deactivated through the growth of contraction of dendritic spines…could be governed by something like the processes involved in quasi-crystal growth. Thus, not just one of the possible alternative arrangements is tried out, but vast numbers, all superposed in complex linear superposition.

From the above, it is deduced that the impact is only on the conscious mind, whereas the unconscious mind is left to do with algorithmic computationality. Why is this important for Penrose is, since, as a mathematician believes in the mathematical ideas as populating an ideal Platonic world, and which in turn is accessible only via the intellect. And harking back to the non-locality principle within quantum theory, it is clear that true intellect requires consciousness, and the mathematician’s conscious mind has a direct route to truth. In the meanwhile, there is a position in “many-worlds” (2) view that supersedes Penrose’s quantum realist one. This position rejects the Reduction Process in favor of Unitary Process, by terming the former as a mere illusion. Penrose shows his reservations against this view, as for him, a theory of consciousness needs to be in place prior to “many-worlds” view, and before the latter view could be squared with what one actually observes. Penrose is quite amazed at how many AI reviewers and researchers embrace the “many-worlds” hypothesis, and mocks at them, for their reasons being better supportive of validating AI project. In short, Penrose’s criticism of strong AI is based on the project’s assertion that consciousness can emerge by a complex system of algorithms, whereas for the thinker, a great many things humans involve in are intrinsically non-algorithmic in nature. For Penrose, a system can be deterministic without being algorithmic. He even uses the Turing’s halting theorem (3) to demonstrate the possibility of replication of consciousness. In a public lecture in Kolkata on the 4th of January 2011 (4), Penrose had this to say,

There are many things in Physics which are yet unknown. Unless we unravel them, we cannot think of creating real artificial intelligence. It cannot be achieved through the present system of computing which is based on mathematical algorithm. We need to be able to replicate human consciousness, which, I believe, is possible through physics and quantum mechanics. The good news is that recent experiments indicate that it is possible.

There is an apparent shift in Penrosean ideas via what he calls “correct quantum gravity”, which argues for the rational processes of the mind to be completely algorithmic and probably standing a bright chance to be duplicated by a sufficiently complex computing system. As he quoted from the same lecture in Kolkata,

A few years back, scientists at NASA had contemplated sending intelligent robots to space and sought my inputs. Even though we are still unable to create some device with genuine feelings and understanding, the question remains a disturbing one. Is it ethical to leave a machine which has consciousness in some faraway planet or in space? Honestly, we haven’t reached that stage yet. Having said that, I must add it may not be too far away, either. It is certainly a possibility.

Penrose does meet up with some sympathizers for his view, but his fellow-travelers do not tread along with him for a long distance. For example, in an interview with Sander Olson, Vernor Vinge, despite showing some reluctance to Penrose’s position, accepts that physical aspects of mind, or especially the quantum effects have not been studied in greater detail, but these quantum effects would simply be another thing to be learned with artifacts. Vinge does speculate on other paradigms that could be equally utilized for AI research hitting speed, rather than confining oneself to computer departments to bank on their progress. His speculations (5) have some parallel to what Penrose and Searle would hint at, albeit occasionally. Most of the work in AI could benefit, if AI, neural nets are closely connected to biological life. Rather than banking upon modeling and understanding of biological life with computers, if composite systems relying on biological life for guidance, or for providing features we do not understand quite well as yet to be implemented within the hardware, could be fathomed and made a reality, the program of AI would undoubtedly push the pedal to accelerate. There would probably be no disagreeing with what Aaron Saenz, Senior Editor of singularityhub.com said (6),

Artificial General Intelligence is one of the Holy Grails of science because it is almost mythical in its promise: not a system that simply learns, but one that reaches and exceeds our own kind of intelligence. A truly new form of advanced life. There are many brilliant people trying to find it. Each of these AI researchers have their own approach, their own expectations and their own history of failures and a precious few successes. The products you see on the market today are narrow AI-machines that have very limited ability to learn. As Scott Brown said, “today’s I technology is so primitive that much of the cleverness goes towards inventing business models that do not require good algorithms to succeed.” We’re in the infantile stages of AI. If that. Maybe the fetal stages.

1398654612801

(1) This is quite apocalyptic sounding like the singularity notion of Ray Kurzweil, which is an extremely disruptive, world-altering event that has the potentiality of forever changing the course of human history. The extermination of humanity by violent machines is not impossible, since there would be no sharp distinctions between men and machines due to the existence of cybernetically enhanced humans and uploaded humans.

(2) “Many-worlds” view was first put forward by Hugh Everett in 1957. According to this view, evolution of state vector regarded realistically, is always governed by deterministic Unitary Process, while Reduction Process remains totally absent from such an evolutionary process. The interesting ramifications of this view are putting conscious observers at the center of the considerations, thus proving the basic assumption that quantum states corresponding to distinct conscious experiences have to be orthogonal (Simon 2009). On the technical side, ‘Orthogonal’ according to quantum mechanics is: two eigenstates of a Hermitian operator, ψm and ψn, are orthogonal if they correspond to different eigenvalues. This means, in Dirac notation, that < ψm | ψn > = 0 unless ψm and ψn correspond to the same eigenvalue. This follows from the fact that Schrödinger’s equation is a Sturm–Liouville equation (in Schrödinger’s formulation) or that observables are given by hermitian operators (in Heisenberg’s formulation).

(3) Halting problem is a decisional problem in computability theory, and is stated as: Given a description of a program, decide whether the program finishes running or continues to run, and will thereby run forever. Turing proved that a general algorithm to solve the halting problem for all possible program-input pairs cannot exist. In a way, the halting problem is undecidable over Turing machines.

(4) Penrose, R. AI may soon become reality. Public lecture delivered in Kolkata on the 4th of Jan 2011. <http://timesofindia.indiatimes.com/city/kolkata-/AI-may-soon-become-reality- Penrose/articleshow/7219700.cms>

(5) Olson, S. Interview with Vernor Vinge in Nanotech.Biz <http://www.nanotech.biz/i.php?id=01_16_09&gt;

(6) Saenz, A. Will Vicarious Systems’ Silicon Valley Pedigree Help it Build AI? in singularityhub.com <http://singularityhub.com/2011/02/03/will-vicarious-systems-silicon-valley-pedigree-help-it-build-agi/&gt;